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This review article attempts to provide a comprehensive review of the recent progress in the so-called resistive random access memories (RRAMs). First, a brief introduction is presented to describe the construction and development of RRAMs, their potential for broad applications in the fields of nonvolatile memory, unconventional computing and logic devices, and the focus of research concerning RRAMs over the past decade. Second, both inorganic and organic materials used in RRAMs are summarized, and their respective advantages and shortcomings are discussed. Third, the important switching mechanisms are discussed in depth and are classified into ion migration, charge trapping/de-trapping, thermochemical reaction, exclusive mechanisms in inorganics, and exclusive mechanisms in organics. Fourth, attention is given to the application of RRAMs for data storage, including their current performance, methods for performance enhancement, sneak-path issue and possible solutions, and demonstrations of 2-D and 3-D crossbar arrays. Fifth, prospective applications of RRAMs in unconventional computing, as well as logic devices and multi-functionalization of RRAMs, are comprehensively summarized and thoroughly discussed. The present review article ends with a short discussion concerning the challenges and future prospects of the RRAMs.